LMFDB - Embedded newform 5850.2.e.e.5149.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5850,2,Mod(5149,5850)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5850, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5850.5149");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5850.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$46.7124851824$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 390)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			5149.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	5850.5149
Dual form			5850.2.e.e.5149.2

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{8} +O(q^{10})$	$q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{8} -4.00000 q^{11} -1.00000i q^{13} +1.00000 q^{16} +6.00000i q^{17} -4.00000 q^{19} +4.00000i q^{22} +8.00000i q^{23} -1.00000 q^{26} +6.00000 q^{29} -8.00000 q^{31} -1.00000i q^{32} +6.00000 q^{34} -10.0000i q^{37} +4.00000i q^{38} +6.00000 q^{41} -4.00000i q^{43} +4.00000 q^{44} +8.00000 q^{46} +7.00000 q^{49} +1.00000i q^{52} -10.0000i q^{53} -6.00000i q^{58} +4.00000 q^{59} -2.00000 q^{61} +8.00000i q^{62} -1.00000 q^{64} -12.0000i q^{67} -6.00000i q^{68} -16.0000 q^{71} -2.00000i q^{73} -10.0000 q^{74} +4.00000 q^{76} +16.0000 q^{79} -6.00000i q^{82} -12.0000i q^{83} -4.00000 q^{86} -4.00000i q^{88} +10.0000 q^{89} -8.00000i q^{92} -6.00000i q^{97} -7.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4}+O(q^{10}) $ 2 * q - 2 * q^4	$ 2 q - 2 q^{4} - 8 q^{11} + 2 q^{16} - 8 q^{19} - 2 q^{26} + 12 q^{29} - 16 q^{31} + 12 q^{34} + 12 q^{41} + 8 q^{44} + 16 q^{46} + 14 q^{49} + 8 q^{59} - 4 q^{61} - 2 q^{64} - 32 q^{71} - 20 q^{74} + 8 q^{76} + 32 q^{79} - 8 q^{86} + 20 q^{89}+O(q^{100}) $ 2 * q - 2 * q^4 - 8 * q^11 + 2 * q^16 - 8 * q^19 - 2 * q^26 + 12 * q^29 - 16 * q^31 + 12 * q^34 + 12 * q^41 + 8 * q^44 + 16 * q^46 + 14 * q^49 + 8 * q^59 - 4 * q^61 - 2 * q^64 - 32 * q^71 - 20 * q^74 + 8 * q^76 + 32 * q^79 - 8 * q^86 + 20 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5850\mathbb{Z}\right)^\times$.

$n$	$2251$	$3251$	$3277$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	1.00000i	0.353553i
$9$	0	0
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	6.00000i	1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$	6.00000i	1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	0	0
$22$	4.00000i	0.852803i
$23$	8.00000i	1.66812i	0.551677	+	0.834058i	$0.313988\pi$
$23$	8.00000i	1.66812i	−0.551677	+	0.834058i	$0.686012\pi$
$24$	0	0
$25$	0	0
$26$	−1.00000	−0.196116
$27$	0	0
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	6.00000	1.02899
$35$	0	0
$36$	0	0
$37$	− 10.0000i	− 1.64399i	−0.569495	−	0.821995i	$-0.692861\pi$
$37$	− 10.0000i	− 1.64399i	0.569495	−	0.821995i	$-0.307139\pi$
$38$	4.00000i	0.648886i
$39$	0	0
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	8.00000	1.17954
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	7.00000	1.00000
$50$	0	0
$51$	0	0
$52$	1.00000i	0.138675i
$53$	− 10.0000i	− 1.37361i	−0.726844	−	0.686803i	$-0.759014\pi$
$53$	− 10.0000i	− 1.37361i	0.726844	−	0.686803i	$-0.240986\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	− 6.00000i	− 0.787839i
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	8.00000i	1.01600i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 12.0000i	− 1.46603i	−0.680211	−	0.733017i	$-0.738112\pi$
$67$	− 12.0000i	− 1.46603i	0.680211	−	0.733017i	$-0.261888\pi$
$68$	− 6.00000i	− 0.727607i
$69$	0	0
$70$	0	0
$71$	−16.0000	−1.89885	−0.949425	−	0.313993i	$-0.898333\pi$
$71$	−16.0000	−1.89885	−0.949425	+	0.313993i	$0.898333\pi$
$72$	0	0
$73$	− 2.00000i	− 0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$	− 2.00000i	− 0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$	−10.0000	−1.16248
$75$	0	0
$76$	4.00000	0.458831
$77$	0	0
$78$	0	0
$79$	16.0000	1.80014	0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000	1.80014	0.900070	+	0.435745i	$0.143515\pi$
$80$	0	0
$81$	0	0
$82$	− 6.00000i	− 0.662589i
$83$	− 12.0000i	− 1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$	− 12.0000i	− 1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$	0	0
$85$	0	0
$86$	−4.00000	−0.431331
$87$	0	0
$88$	− 4.00000i	− 0.426401i
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	0	0
$92$	− 8.00000i	− 0.834058i
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 6.00000i	− 0.609208i	−0.952479	−	0.304604i	$-0.901476\pi$
$97$	− 6.00000i	− 0.609208i	0.952479	−	0.304604i	$-0.0985241\pi$
$98$	− 7.00000i	− 0.707107i
$99$	0	0
$100$	0	0
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	−10.0000	−0.971286
$107$	12.0000i	1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$	12.0000i	1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	10.0000i	0.940721i	0.882474	+	0.470360i	$0.155876\pi$
$113$	10.0000i	0.940721i	−0.882474	+	0.470360i	$0.844124\pi$
$114$	0	0
$115$	0	0
$116$	−6.00000	−0.557086
$117$	0	0
$118$	− 4.00000i	− 0.368230i
$119$	0	0
$120$	0	0
$121$	5.00000	0.454545
$122$	2.00000i	0.181071i
$123$	0	0
$124$	8.00000	0.718421
$125$	0	0
$126$	0	0
$127$	8.00000i	0.709885i	0.934888	+	0.354943i	$0.115500\pi$
$127$	8.00000i	0.709885i	−0.934888	+	0.354943i	$0.884500\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	0	0
$134$	−12.0000	−1.03664
$135$	0	0
$136$	−6.00000	−0.514496
$137$	− 10.0000i	− 0.854358i	−0.904167	−	0.427179i	$-0.859507\pi$
$137$	− 10.0000i	− 0.854358i	0.904167	−	0.427179i	$-0.140493\pi$
$138$	0	0
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	0	0
$141$	0	0
$142$	16.0000i	1.34269i
$143$	4.00000i	0.334497i
$144$	0	0
$145$	0	0
$146$	−2.00000	−0.165521
$147$	0	0
$148$	10.0000i	0.821995i
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	− 4.00000i	− 0.324443i
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 2.00000i	− 0.159617i	−0.996810	−	0.0798087i	$-0.974569\pi$
$157$	− 2.00000i	− 0.159617i	0.996810	−	0.0798087i	$-0.0254309\pi$
$158$	− 16.0000i	− 1.27289i
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 4.00000i	− 0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$	− 4.00000i	− 0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	−12.0000	−0.931381
$167$	− 8.00000i	− 0.619059i	−0.950890	−	0.309529i	$-0.899829\pi$
$167$	− 8.00000i	− 0.619059i	0.950890	−	0.309529i	$-0.100171\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	4.00000i	0.304997i
$173$	− 18.0000i	− 1.36851i	−0.729241	−	0.684257i	$-0.760127\pi$
$173$	− 18.0000i	− 1.36851i	0.729241	−	0.684257i	$-0.239873\pi$
$174$	0	0
$175$	0	0
$176$	−4.00000	−0.301511
$177$	0	0
$178$	− 10.0000i	− 0.749532i
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	0	0
$183$	0	0
$184$	−8.00000	−0.589768
$185$	0	0
$186$	0	0
$187$	− 24.0000i	− 1.75505i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	− 10.0000i	− 0.719816i	−0.932988	−	0.359908i	$-0.882808\pi$
$193$	− 10.0000i	− 0.719816i	0.932988	−	0.359908i	$-0.117192\pi$
$194$	−6.00000	−0.430775
$195$	0	0
$196$	−7.00000	−0.500000
$197$	− 6.00000i	− 0.427482i	−0.976890	−	0.213741i	$-0.931435\pi$
$197$	− 6.00000i	− 0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	0	0
$202$	− 2.00000i	− 0.140720i
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	− 1.00000i	− 0.0693375i
$209$	16.0000	1.10674
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	10.0000i	0.686803i
$213$	0	0
$214$	12.0000	0.820303
$215$	0	0
$216$	0	0
$217$	0	0
$218$	− 10.0000i	− 0.677285i
$219$	0	0
$220$	0	0
$221$	6.00000	0.403604
$222$	0	0
$223$	− 8.00000i	− 0.535720i	−0.963458	−	0.267860i	$-0.913684\pi$
$223$	− 8.00000i	− 0.535720i	0.963458	−	0.267860i	$-0.0863164\pi$
$224$	0	0
$225$	0	0
$226$	10.0000	0.665190
$227$	− 4.00000i	− 0.265489i	−0.991150	−	0.132745i	$-0.957621\pi$
$227$	− 4.00000i	− 0.265489i	0.991150	−	0.132745i	$-0.0423790\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	0	0
$232$	6.00000i	0.393919i
$233$	− 14.0000i	− 0.917170i	−0.888650	−	0.458585i	$-0.848356\pi$
$233$	− 14.0000i	− 0.917170i	0.888650	−	0.458585i	$-0.151644\pi$
$234$	0	0
$235$	0	0
$236$	−4.00000	−0.260378
$237$	0	0
$238$	0	0
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	0	0
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	− 5.00000i	− 0.321412i
$243$	0	0
$244$	2.00000	0.128037
$245$	0	0
$246$	0	0
$247$	4.00000i	0.254514i
$248$	− 8.00000i	− 0.508001i
$249$	0	0
$250$	0	0
$251$	28.0000	1.76734	0.883672	−	0.468106i	$-0.155064\pi$
$251$	28.0000	1.76734	0.883672	+	0.468106i	$0.155064\pi$
$252$	0	0
$253$	− 32.0000i	− 2.01182i
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	6.00000i	0.374270i	0.982334	+	0.187135i	$0.0599201\pi$
$257$	6.00000i	0.374270i	−0.982334	+	0.187135i	$0.940080\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	12.0000i	0.741362i
$263$	24.0000i	1.47990i	0.672660	+	0.739952i	$0.265152\pi$
$263$	24.0000i	1.47990i	−0.672660	+	0.739952i	$0.734848\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	12.0000i	0.733017i
$269$	−26.0000	−1.58525	−0.792624	−	0.609711i	$-0.791286\pi$
$269$	−26.0000	−1.58525	−0.792624	+	0.609711i	$0.791286\pi$
$270$	0	0
$271$	24.0000	1.45790	0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000	1.45790	0.728948	+	0.684569i	$0.240010\pi$
$272$	6.00000i	0.363803i
$273$	0	0
$274$	−10.0000	−0.604122
$275$	0	0
$276$	0	0
$277$	− 10.0000i	− 0.600842i	−0.953807	−	0.300421i	$-0.902873\pi$
$277$	− 10.0000i	− 0.600842i	0.953807	−	0.300421i	$-0.0971271\pi$
$278$	12.0000i	0.719712i
$279$	0	0
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	− 20.0000i	− 1.18888i	−0.804141	−	0.594438i	$-0.797374\pi$
$283$	− 20.0000i	− 1.18888i	0.804141	−	0.594438i	$-0.202626\pi$
$284$	16.0000	0.949425
$285$	0	0
$286$	4.00000	0.236525
$287$	0	0
$288$	0	0
$289$	−19.0000	−1.11765
$290$	0	0
$291$	0	0
$292$	2.00000i	0.117041i
$293$	− 26.0000i	− 1.51894i	−0.650545	−	0.759468i	$-0.725459\pi$
$293$	− 26.0000i	− 1.51894i	0.650545	−	0.759468i	$-0.274541\pi$
$294$	0	0
$295$	0	0
$296$	10.0000	0.581238
$297$	0	0
$298$	− 6.00000i	− 0.347571i
$299$	8.00000	0.462652
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	−4.00000	−0.229416
$305$	0	0
$306$	0	0
$307$	− 12.0000i	− 0.684876i	−0.939540	−	0.342438i	$-0.888747\pi$
$307$	− 12.0000i	− 0.684876i	0.939540	−	0.342438i	$-0.111253\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	22.0000i	1.24351i	0.783210	+	0.621757i	$0.213581\pi$
$313$	22.0000i	1.24351i	−0.783210	+	0.621757i	$0.786419\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	−16.0000	−0.900070
$317$	2.00000i	0.112331i	0.998421	+	0.0561656i	$0.0178875\pi$
$317$	2.00000i	0.112331i	−0.998421	+	0.0561656i	$0.982113\pi$
$318$	0	0
$319$	−24.0000	−1.34374
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 24.0000i	− 1.33540i
$324$	0	0
$325$	0	0
$326$	−4.00000	−0.221540
$327$	0	0
$328$	6.00000i	0.331295i
$329$	0	0
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	12.0000i	0.658586i
$333$	0	0
$334$	−8.00000	−0.437741
$335$	0	0
$336$	0	0
$337$	2.00000i	0.108947i	0.998515	+	0.0544735i	$0.0173480\pi$
$337$	2.00000i	0.108947i	−0.998515	+	0.0544735i	$0.982652\pi$
$338$	1.00000i	0.0543928i
$339$	0	0
$340$	0	0
$341$	32.0000	1.73290
$342$	0	0
$343$	0	0
$344$	4.00000	0.215666
$345$	0	0
$346$	−18.0000	−0.967686
$347$	− 20.0000i	− 1.07366i	−0.843692	−	0.536828i	$-0.819622\pi$
$347$	− 20.0000i	− 1.07366i	0.843692	−	0.536828i	$-0.180378\pi$
$348$	0	0
$349$	−6.00000	−0.321173	−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000	−0.321173	−0.160586	+	0.987022i	$0.551338\pi$
$350$	0	0
$351$	0	0
$352$	4.00000i	0.213201i
$353$	− 30.0000i	− 1.59674i	−0.602168	−	0.798369i	$-0.705696\pi$
$353$	− 30.0000i	− 1.59674i	0.602168	−	0.798369i	$-0.294304\pi$
$354$	0	0
$355$	0	0
$356$	−10.0000	−0.529999
$357$	0	0
$358$	− 12.0000i	− 0.634220i
$359$	16.0000	0.844448	0.422224	−	0.906492i	$-0.361250\pi$
$359$	16.0000	0.844448	0.422224	+	0.906492i	$0.361250\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	− 6.00000i	− 0.315353i
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 24.0000i	− 1.25279i	−0.779506	−	0.626395i	$-0.784530\pi$
$367$	− 24.0000i	− 1.25279i	0.779506	−	0.626395i	$-0.215470\pi$
$368$	8.00000i	0.417029i
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 22.0000i	− 1.13912i	−0.821951	−	0.569558i	$-0.807114\pi$
$373$	− 22.0000i	− 1.13912i	0.821951	−	0.569558i	$-0.192886\pi$
$374$	−24.0000	−1.24101
$375$	0	0
$376$	0	0
$377$	− 6.00000i	− 0.309016i
$378$	0	0
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	32.0000i	1.63512i	0.575841	+	0.817562i	$0.304675\pi$
$383$	32.0000i	1.63512i	−0.575841	+	0.817562i	$0.695325\pi$
$384$	0	0
$385$	0	0
$386$	−10.0000	−0.508987
$387$	0	0
$388$	6.00000i	0.304604i
$389$	−34.0000	−1.72387	−0.861934	−	0.507020i	$-0.830747\pi$
$389$	−34.0000	−1.72387	−0.861934	+	0.507020i	$0.830747\pi$
$390$	0	0
$391$	−48.0000	−2.42746
$392$	7.00000i	0.353553i
$393$	0	0
$394$	−6.00000	−0.302276
$395$	0	0
$396$	0	0
$397$	30.0000i	1.50566i	0.658217	+	0.752828i	$0.271311\pi$
$397$	30.0000i	1.50566i	−0.658217	+	0.752828i	$0.728689\pi$
$398$	− 8.00000i	− 0.401004i
$399$	0	0
$400$	0	0
$401$	−34.0000	−1.69788	−0.848939	−	0.528490i	$-0.822758\pi$
$401$	−34.0000	−1.69788	−0.848939	+	0.528490i	$0.822758\pi$
$402$	0	0
$403$	8.00000i	0.398508i
$404$	−2.00000	−0.0995037
$405$	0	0
$406$	0	0
$407$	40.0000i	1.98273i
$408$	0	0
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−1.00000	−0.0490290
$417$	0	0
$418$	− 16.0000i	− 0.782586i
$419$	−4.00000	−0.195413	−0.0977064	−	0.995215i	$-0.531151\pi$
$419$	−4.00000	−0.195413	−0.0977064	+	0.995215i	$0.531151\pi$
$420$	0	0
$421$	−34.0000	−1.65706	−0.828529	−	0.559946i	$-0.810822\pi$
$421$	−34.0000	−1.65706	−0.828529	+	0.559946i	$0.810822\pi$
$422$	12.0000i	0.584151i
$423$	0	0
$424$	10.0000	0.485643
$425$	0	0
$426$	0	0
$427$	0	0
$428$	− 12.0000i	− 0.580042i
$429$	0	0
$430$	0	0
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	0	0
$433$	− 34.0000i	− 1.63394i	−0.576683	−	0.816968i	$-0.695653\pi$
$433$	− 34.0000i	− 1.63394i	0.576683	−	0.816968i	$-0.304347\pi$
$434$	0	0
$435$	0	0
$436$	−10.0000	−0.478913
$437$	− 32.0000i	− 1.53077i
$438$	0	0
$439$	40.0000	1.90910	0.954548	−	0.298057i	$-0.0963387\pi$
$439$	40.0000	1.90910	0.954548	+	0.298057i	$0.0963387\pi$
$440$	0	0
$441$	0	0
$442$	− 6.00000i	− 0.285391i
$443$	36.0000i	1.71041i	0.518289	+	0.855206i	$0.326569\pi$
$443$	36.0000i	1.71041i	−0.518289	+	0.855206i	$0.673431\pi$
$444$	0	0
$445$	0	0
$446$	−8.00000	−0.378811
$447$	0	0
$448$	0	0
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	0	0
$451$	−24.0000	−1.13012
$452$	− 10.0000i	− 0.470360i
$453$	0	0
$454$	−4.00000	−0.187729
$455$	0	0
$456$	0	0
$457$	34.0000i	1.59045i	0.606313	+	0.795226i	$0.292648\pi$
$457$	34.0000i	1.59045i	−0.606313	+	0.795226i	$0.707352\pi$
$458$	14.0000i	0.654177i
$459$	0	0
$460$	0	0
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	0	0
$463$	− 8.00000i	− 0.371792i	−0.982569	−	0.185896i	$-0.940481\pi$
$463$	− 8.00000i	− 0.371792i	0.982569	−	0.185896i	$-0.0595187\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	−14.0000	−0.648537
$467$	− 12.0000i	− 0.555294i	−0.960683	−	0.277647i	$-0.910445\pi$
$467$	− 12.0000i	− 0.555294i	0.960683	−	0.277647i	$-0.0895545\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	4.00000i	0.184115i
$473$	16.0000i	0.735681i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	8.00000i	0.365911i
$479$	−8.00000	−0.365529	−0.182765	−	0.983157i	$-0.558505\pi$
$479$	−8.00000	−0.365529	−0.182765	+	0.983157i	$0.558505\pi$
$480$	0	0
$481$	−10.0000	−0.455961
$482$	− 18.0000i	− 0.819878i
$483$	0	0
$484$	−5.00000	−0.227273
$485$	0	0
$486$	0	0
$487$	− 16.0000i	− 0.725029i	−0.931978	−	0.362515i	$-0.881918\pi$
$487$	− 16.0000i	− 0.725029i	0.931978	−	0.362515i	$-0.118082\pi$
$488$	− 2.00000i	− 0.0905357i
$489$	0	0
$490$	0	0
$491$	−4.00000	−0.180517	−0.0902587	−	0.995918i	$-0.528769\pi$
$491$	−4.00000	−0.180517	−0.0902587	+	0.995918i	$0.528769\pi$
$492$	0	0
$493$	36.0000i	1.62136i
$494$	4.00000	0.179969
$495$	0	0
$496$	−8.00000	−0.359211
$497$	0	0
$498$	0	0
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	0	0
$501$	0	0
$502$	− 28.0000i	− 1.24970i
$503$	8.00000i	0.356702i	0.983967	+	0.178351i	$0.0570763\pi$
$503$	8.00000i	0.356702i	−0.983967	+	0.178351i	$0.942924\pi$
$504$	0	0
$505$	0	0
$506$	−32.0000	−1.42257
$507$	0	0
$508$	− 8.00000i	− 0.354943i
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	0	0
$511$	0	0
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	6.00000	0.264649
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	0	0
$523$	− 36.0000i	− 1.57417i	−0.616844	−	0.787085i	$-0.711589\pi$
$523$	− 36.0000i	− 1.57417i	0.616844	−	0.787085i	$-0.288411\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	24.0000	1.04645
$527$	− 48.0000i	− 2.09091i
$528$	0	0
$529$	−41.0000	−1.78261
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 6.00000i	− 0.259889i
$534$	0	0
$535$	0	0
$536$	12.0000	0.518321
$537$	0	0
$538$	26.0000i	1.12094i
$539$	−28.0000	−1.20605
$540$	0	0
$541$	22.0000	0.945854	0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000	0.945854	0.472927	+	0.881102i	$0.343197\pi$
$542$	− 24.0000i	− 1.03089i
$543$	0	0
$544$	6.00000	0.257248
$545$	0	0
$546$	0	0
$547$	− 20.0000i	− 0.855138i	−0.903983	−	0.427569i	$-0.859370\pi$
$547$	− 20.0000i	− 0.855138i	0.903983	−	0.427569i	$-0.140630\pi$
$548$	10.0000i	0.427179i
$549$	0	0
$550$	0	0
$551$	−24.0000	−1.02243
$552$	0	0
$553$	0	0
$554$	−10.0000	−0.424859
$555$	0	0
$556$	12.0000	0.508913
$557$	18.0000i	0.762684i	0.924434	+	0.381342i	$0.124538\pi$
$557$	18.0000i	0.762684i	−0.924434	+	0.381342i	$0.875462\pi$
$558$	0	0
$559$	−4.00000	−0.169182
$560$	0	0
$561$	0	0
$562$	10.0000i	0.421825i
$563$	12.0000i	0.505740i	0.967500	+	0.252870i	$0.0813744\pi$
$563$	12.0000i	0.505740i	−0.967500	+	0.252870i	$0.918626\pi$
$564$	0	0
$565$	0	0
$566$	−20.0000	−0.840663
$567$	0	0
$568$	− 16.0000i	− 0.671345i
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	0	0
$571$	28.0000	1.17176	0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000	1.17176	0.585882	+	0.810397i	$0.300748\pi$
$572$	− 4.00000i	− 0.167248i
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	42.0000i	1.74848i	0.485491	+	0.874241i	$0.338641\pi$
$577$	42.0000i	1.74848i	−0.485491	+	0.874241i	$0.661359\pi$
$578$	19.0000i	0.790296i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	40.0000i	1.65663i
$584$	2.00000	0.0827606
$585$	0	0
$586$	−26.0000	−1.07405
$587$	− 12.0000i	− 0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$	− 12.0000i	− 0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$	0	0
$589$	32.0000	1.31854
$590$	0	0
$591$	0	0
$592$	− 10.0000i	− 0.410997i
$593$	34.0000i	1.39621i	0.715994	+	0.698106i	$0.245974\pi$
$593$	34.0000i	1.39621i	−0.715994	+	0.698106i	$0.754026\pi$
$594$	0	0
$595$	0	0
$596$	−6.00000	−0.245770
$597$	0	0
$598$	− 8.00000i	− 0.327144i
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	−6.00000	−0.244745	−0.122373	−	0.992484i	$-0.539050\pi$
$601$	−6.00000	−0.244745	−0.122373	+	0.992484i	$0.539050\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 8.00000i	− 0.324710i	−0.986732	−	0.162355i	$-0.948091\pi$
$607$	− 8.00000i	− 0.324710i	0.986732	−	0.162355i	$-0.0519090\pi$
$608$	4.00000i	0.162221i
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	10.0000i	0.403896i	0.979396	+	0.201948i	$0.0647272\pi$
$613$	10.0000i	0.403896i	−0.979396	+	0.201948i	$0.935273\pi$
$614$	−12.0000	−0.484281
$615$	0	0
$616$	0	0
$617$	− 42.0000i	− 1.69086i	−0.534089	−	0.845428i	$-0.679345\pi$
$617$	− 42.0000i	− 1.69086i	0.534089	−	0.845428i	$-0.320655\pi$
$618$	0	0
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	0	0
$621$	0	0
$622$	− 24.0000i	− 0.962312i
$623$	0	0
$624$	0	0
$625$	0	0
$626$	22.0000	0.879297
$627$	0	0
$628$	2.00000i	0.0798087i
$629$	60.0000	2.39236
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	16.0000i	0.636446i
$633$	0	0
$634$	2.00000	0.0794301
$635$	0	0
$636$	0	0
$637$	− 7.00000i	− 0.277350i
$638$	24.0000i	0.950169i
$639$	0	0
$640$	0	0
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	0	0
$643$	− 4.00000i	− 0.157745i	−0.996885	−	0.0788723i	$-0.974868\pi$
$643$	− 4.00000i	− 0.157745i	0.996885	−	0.0788723i	$-0.0251319\pi$
$644$	0	0
$645$	0	0
$646$	−24.0000	−0.944267
$647$	− 24.0000i	− 0.943537i	−0.881722	−	0.471769i	$-0.843616\pi$
$647$	− 24.0000i	− 0.943537i	0.881722	−	0.471769i	$-0.156384\pi$
$648$	0	0
$649$	−16.0000	−0.628055
$650$	0	0
$651$	0	0
$652$	4.00000i	0.156652i
$653$	− 34.0000i	− 1.33052i	−0.746611	−	0.665261i	$-0.768320\pi$
$653$	− 34.0000i	− 1.33052i	0.746611	−	0.665261i	$-0.231680\pi$
$654$	0	0
$655$	0	0
$656$	6.00000	0.234261
$657$	0	0
$658$	0	0
$659$	−20.0000	−0.779089	−0.389545	−	0.921008i	$-0.627368\pi$
$659$	−20.0000	−0.779089	−0.389545	+	0.921008i	$0.627368\pi$
$660$	0	0
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	20.0000i	0.777322i
$663$	0	0
$664$	12.0000	0.465690
$665$	0	0
$666$	0	0
$667$	48.0000i	1.85857i
$668$	8.00000i	0.309529i
$669$	0	0
$670$	0	0
$671$	8.00000	0.308837
$672$	0	0
$673$	− 34.0000i	− 1.31060i	−0.755367	−	0.655302i	$-0.772541\pi$
$673$	− 34.0000i	− 1.31060i	0.755367	−	0.655302i	$-0.227459\pi$
$674$	2.00000	0.0770371
$675$	0	0
$676$	1.00000	0.0384615
$677$	− 22.0000i	− 0.845529i	−0.906240	−	0.422764i	$-0.861060\pi$
$677$	− 22.0000i	− 0.845529i	0.906240	−	0.422764i	$-0.138940\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	− 32.0000i	− 1.22534i
$683$	28.0000i	1.07139i	0.844411	+	0.535695i	$0.179950\pi$
$683$	28.0000i	1.07139i	−0.844411	+	0.535695i	$0.820050\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	− 4.00000i	− 0.152499i
$689$	−10.0000	−0.380970
$690$	0	0
$691$	−12.0000	−0.456502	−0.228251	−	0.973602i	$-0.573301\pi$
$691$	−12.0000	−0.456502	−0.228251	+	0.973602i	$0.573301\pi$
$692$	18.0000i	0.684257i
$693$	0	0
$694$	−20.0000	−0.759190
$695$	0	0
$696$	0	0
$697$	36.0000i	1.36360i
$698$	6.00000i	0.227103i
$699$	0	0
$700$	0	0
$701$	−22.0000	−0.830929	−0.415464	−	0.909610i	$-0.636381\pi$
$701$	−22.0000	−0.830929	−0.415464	+	0.909610i	$0.636381\pi$
$702$	0	0
$703$	40.0000i	1.50863i
$704$	4.00000	0.150756
$705$	0	0
$706$	−30.0000	−1.12906
$707$	0	0
$708$	0	0
$709$	50.0000	1.87779	0.938895	−	0.344204i	$-0.111851\pi$
$709$	50.0000	1.87779	0.938895	+	0.344204i	$0.111851\pi$
$710$	0	0
$711$	0	0
$712$	10.0000i	0.374766i
$713$	− 64.0000i	− 2.39682i
$714$	0	0
$715$	0	0
$716$	−12.0000	−0.448461
$717$	0	0
$718$	− 16.0000i	− 0.597115i
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	3.00000i	0.111648i
$723$	0	0
$724$	−6.00000	−0.222988
$725$	0	0
$726$	0	0
$727$	− 16.0000i	− 0.593407i	−0.954970	−	0.296704i	$-0.904113\pi$
$727$	− 16.0000i	− 0.593407i	0.954970	−	0.296704i	$-0.0958873\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	24.0000	0.887672
$732$	0	0
$733$	50.0000i	1.84679i	0.383849	+	0.923396i	$0.374598\pi$
$733$	50.0000i	1.84679i	−0.383849	+	0.923396i	$0.625402\pi$
$734$	−24.0000	−0.885856
$735$	0	0
$736$	8.00000	0.294884
$737$	48.0000i	1.76810i
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 24.0000i	− 0.880475i	−0.897881	−	0.440237i	$-0.854894\pi$
$743$	− 24.0000i	− 0.880475i	0.897881	−	0.440237i	$-0.145106\pi$
$744$	0	0
$745$	0	0
$746$	−22.0000	−0.805477
$747$	0	0
$748$	24.0000i	0.877527i
$749$	0	0
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	0	0
$753$	0	0
$754$	−6.00000	−0.218507
$755$	0	0
$756$	0	0
$757$	− 42.0000i	− 1.52652i	−0.646094	−	0.763258i	$-0.723599\pi$
$757$	− 42.0000i	− 1.52652i	0.646094	−	0.763258i	$-0.276401\pi$
$758$	− 4.00000i	− 0.145287i
$759$	0	0
$760$	0	0
$761$	−42.0000	−1.52250	−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000	−1.52250	−0.761249	+	0.648459i	$0.775414\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	32.0000	1.15621
$767$	− 4.00000i	− 0.144432i
$768$	0	0
$769$	−34.0000	−1.22607	−0.613036	−	0.790055i	$-0.710052\pi$
$769$	−34.0000	−1.22607	−0.613036	+	0.790055i	$0.710052\pi$
$770$	0	0
$771$	0	0
$772$	10.0000i	0.359908i
$773$	38.0000i	1.36677i	0.730061	+	0.683383i	$0.239492\pi$
$773$	38.0000i	1.36677i	−0.730061	+	0.683383i	$0.760508\pi$
$774$	0	0
$775$	0	0
$776$	6.00000	0.215387
$777$	0	0
$778$	34.0000i	1.21896i
$779$	−24.0000	−0.859889
$780$	0	0
$781$	64.0000	2.29010
$782$	48.0000i	1.71648i
$783$	0	0
$784$	7.00000	0.250000
$785$	0	0
$786$	0	0
$787$	4.00000i	0.142585i	0.997455	+	0.0712923i	$0.0227123\pi$
$787$	4.00000i	0.142585i	−0.997455	+	0.0712923i	$0.977288\pi$
$788$	6.00000i	0.213741i
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	2.00000i	0.0710221i
$794$	30.0000	1.06466
$795$	0	0
$796$	−8.00000	−0.283552
$797$	2.00000i	0.0708436i	0.999372	+	0.0354218i	$0.0112775\pi$
$797$	2.00000i	0.0708436i	−0.999372	+	0.0354218i	$0.988723\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	34.0000i	1.20058i
$803$	8.00000i	0.282314i
$804$	0	0
$805$	0	0
$806$	8.00000	0.281788
$807$	0	0
$808$	2.00000i	0.0703598i
$809$	42.0000	1.47664	0.738321	−	0.674450i	$-0.235619\pi$
$809$	42.0000	1.47664	0.738321	+	0.674450i	$0.235619\pi$
$810$	0	0
$811$	12.0000	0.421377	0.210688	−	0.977553i	$-0.432429\pi$
$811$	12.0000	0.421377	0.210688	+	0.977553i	$0.432429\pi$
$812$	0	0
$813$	0	0
$814$	40.0000	1.40200
$815$	0	0
$816$	0	0
$817$	16.0000i	0.559769i
$818$	− 22.0000i	− 0.769212i
$819$	0	0
$820$	0	0
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	0	0
$823$	− 16.0000i	− 0.557725i	−0.960331	−	0.278862i	$-0.910043\pi$
$823$	− 16.0000i	− 0.557725i	0.960331	−	0.278862i	$-0.0899574\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	36.0000i	1.25184i	0.779886	+	0.625921i	$0.215277\pi$
$827$	36.0000i	1.25184i	−0.779886	+	0.625921i	$0.784723\pi$
$828$	0	0
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	0	0
$831$	0	0
$832$	1.00000i	0.0346688i
$833$	42.0000i	1.45521i
$834$	0	0
$835$	0	0
$836$	−16.0000	−0.553372
$837$	0	0
$838$	4.00000i	0.138178i
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	34.0000i	1.17172i
$843$	0	0
$844$	12.0000	0.413057
$845$	0	0
$846$	0	0
$847$	0	0
$848$	− 10.0000i	− 0.343401i
$849$	0	0
$850$	0	0
$851$	80.0000	2.74236
$852$	0	0
$853$	26.0000i	0.890223i	0.895475	+	0.445112i	$0.146836\pi$
$853$	26.0000i	0.890223i	−0.895475	+	0.445112i	$0.853164\pi$
$854$	0	0
$855$	0	0
$856$	−12.0000	−0.410152
$857$	− 18.0000i	− 0.614868i	−0.951569	−	0.307434i	$-0.900530\pi$
$857$	− 18.0000i	− 0.614868i	0.951569	−	0.307434i	$-0.0994704\pi$
$858$	0	0
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	0	0
$861$	0	0
$862$	− 24.0000i	− 0.817443i
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	0	0
$866$	−34.0000	−1.15537
$867$	0	0
$868$	0	0
$869$	−64.0000	−2.17105
$870$	0	0
$871$	−12.0000	−0.406604
$872$	10.0000i	0.338643i
$873$	0	0
$874$	−32.0000	−1.08242
$875$	0	0
$876$	0	0
$877$	14.0000i	0.472746i	0.971662	+	0.236373i	$0.0759588\pi$
$877$	14.0000i	0.472746i	−0.971662	+	0.236373i	$0.924041\pi$
$878$	− 40.0000i	− 1.34993i
$879$	0	0
$880$	0	0
$881$	−2.00000	−0.0673817	−0.0336909	−	0.999432i	$-0.510726\pi$
$881$	−2.00000	−0.0673817	−0.0336909	+	0.999432i	$0.510726\pi$
$882$	0	0
$883$	36.0000i	1.21150i	0.795656	+	0.605748i	$0.207126\pi$
$883$	36.0000i	1.21150i	−0.795656	+	0.605748i	$0.792874\pi$
$884$	−6.00000	−0.201802
$885$	0	0
$886$	36.0000	1.20944
$887$	− 8.00000i	− 0.268614i	−0.990940	−	0.134307i	$-0.957119\pi$
$887$	− 8.00000i	− 0.268614i	0.990940	−	0.134307i	$-0.0428808\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	8.00000i	0.267860i
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	− 18.0000i	− 0.600668i
$899$	−48.0000	−1.60089
$900$	0	0
$901$	60.0000	1.99889
$902$	24.0000i	0.799113i
$903$	0	0
$904$	−10.0000	−0.332595
$905$	0	0
$906$	0	0
$907$	− 12.0000i	− 0.398453i	−0.979953	−	0.199227i	$-0.936157\pi$
$907$	− 12.0000i	− 0.398453i	0.979953	−	0.199227i	$-0.0638430\pi$
$908$	4.00000i	0.132745i
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	48.0000i	1.58857i
$914$	34.0000	1.12462
$915$	0	0
$916$	14.0000	0.462573
$917$	0	0
$918$	0	0
$919$	56.0000	1.84727	0.923635	−	0.383274i	$-0.125203\pi$
$919$	56.0000	1.84727	0.923635	+	0.383274i	$0.125203\pi$
$920$	0	0
$921$	0	0
$922$	− 18.0000i	− 0.592798i
$923$	16.0000i	0.526646i
$924$	0	0
$925$	0	0
$926$	−8.00000	−0.262896
$927$	0	0
$928$	− 6.00000i	− 0.196960i
$929$	−14.0000	−0.459325	−0.229663	−	0.973270i	$-0.573762\pi$
$929$	−14.0000	−0.459325	−0.229663	+	0.973270i	$0.573762\pi$
$930$	0	0
$931$	−28.0000	−0.917663
$932$	14.0000i	0.458585i
$933$	0	0
$934$	−12.0000	−0.392652
$935$	0	0
$936$	0	0
$937$	− 22.0000i	− 0.718709i	−0.933201	−	0.359354i	$-0.882997\pi$
$937$	− 22.0000i	− 0.718709i	0.933201	−	0.359354i	$-0.117003\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−30.0000	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$942$	0	0
$943$	48.0000i	1.56310i
$944$	4.00000	0.130189
$945$	0	0
$946$	16.0000	0.520205
$947$	− 52.0000i	− 1.68977i	−0.534946	−	0.844886i	$-0.679668\pi$
$947$	− 52.0000i	− 1.68977i	0.534946	−	0.844886i	$-0.320332\pi$
$948$	0	0
$949$	−2.00000	−0.0649227
$950$	0	0
$951$	0	0
$952$	0	0
$953$	34.0000i	1.10137i	0.834714	+	0.550684i	$0.185633\pi$
$953$	34.0000i	1.10137i	−0.834714	+	0.550684i	$0.814367\pi$
$954$	0	0
$955$	0	0
$956$	8.00000	0.258738
$957$	0	0
$958$	8.00000i	0.258468i
$959$	0	0
$960$	0	0
$961$	33.0000	1.06452
$962$	10.0000i	0.322413i
$963$	0	0
$964$	−18.0000	−0.579741
$965$	0	0
$966$	0	0
$967$	16.0000i	0.514525i	0.966342	+	0.257263i	$0.0828206\pi$
$967$	16.0000i	0.514525i	−0.966342	+	0.257263i	$0.917179\pi$
$968$	5.00000i	0.160706i
$969$	0	0
$970$	0	0
$971$	−36.0000	−1.15529	−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000	−1.15529	−0.577647	+	0.816286i	$0.696029\pi$
$972$	0	0
$973$	0	0
$974$	−16.0000	−0.512673
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	30.0000i	0.959785i	0.877327	+	0.479893i	$0.159324\pi$
$977$	30.0000i	0.959785i	−0.877327	+	0.479893i	$0.840676\pi$
$978$	0	0
$979$	−40.0000	−1.27841
$980$	0	0
$981$	0	0
$982$	4.00000i	0.127645i
$983$	8.00000i	0.255160i	0.991828	+	0.127580i	$0.0407210\pi$
$983$	8.00000i	0.255160i	−0.991828	+	0.127580i	$0.959279\pi$
$984$	0	0
$985$	0	0
$986$	36.0000	1.14647
$987$	0	0
$988$	− 4.00000i	− 0.127257i
$989$	32.0000	1.01754
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	8.00000i	0.254000i
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 58.0000i	− 1.83688i	−0.395562	−	0.918439i	$-0.629450\pi$
$997$	− 58.0000i	− 1.83688i	0.395562	−	0.918439i	$-0.370550\pi$
$998$	20.0000i	0.633089i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5850.2.e.e.5149.1		2
3.2	odd	2		1950.2.e.g.1249.2		2
5.2	odd	4		5850.2.a.bo.1.1		1
5.3	odd	4		1170.2.a.a.1.1		1
5.4	even	2	inner	5850.2.e.e.5149.2		2
15.2	even	4		1950.2.a.k.1.1		1
15.8	even	4		390.2.a.f.1.1	✓	1
15.14	odd	2		1950.2.e.g.1249.1		2
20.3	even	4		9360.2.a.p.1.1		1
60.23	odd	4		3120.2.a.w.1.1		1
195.8	odd	4		5070.2.b.d.1351.2		2
195.38	even	4		5070.2.a.a.1.1		1
195.83	odd	4		5070.2.b.d.1351.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.a.f.1.1	✓	1	15.8	even	4
1170.2.a.a.1.1		1	5.3	odd	4
1950.2.a.k.1.1		1	15.2	even	4
1950.2.e.g.1249.1		2	15.14	odd	2
1950.2.e.g.1249.2		2	3.2	odd	2
3120.2.a.w.1.1		1	60.23	odd	4
5070.2.a.a.1.1		1	195.38	even	4
5070.2.b.d.1351.1		2	195.83	odd	4
5070.2.b.d.1351.2		2	195.8	odd	4
5850.2.a.bo.1.1		1	5.2	odd	4
5850.2.e.e.5149.1		2	1.1	even	1	trivial
5850.2.e.e.5149.2		2	5.4	even	2	inner
9360.2.a.p.1.1		1	20.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5850.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{8} -4.00000 q^{11} -1.00000i q^{13} +1.00000 q^{16} +6.00000i q^{17} -4.00000 q^{19} +4.00000i q^{22} +8.00000i q^{23} -1.00000 q^{26} +6.00000 q^{29} -8.00000 q^{31} -1.00000i q^{32} +6.00000 q^{34} -10.0000i q^{37} +4.00000i q^{38} +6.00000 q^{41} -4.00000i q^{43} +4.00000 q^{44} +8.00000 q^{46} +7.00000 q^{49} +1.00000i q^{52} -10.0000i q^{53} -6.00000i q^{58} +4.00000 q^{59} -2.00000 q^{61} +8.00000i q^{62} -1.00000 q^{64} -12.0000i q^{67} -6.00000i q^{68} -16.0000 q^{71} -2.00000i q^{73} -10.0000 q^{74} +4.00000 q^{76} +16.0000 q^{79} -6.00000i q^{82} -12.0000i q^{83} -4.00000 q^{86} -4.00000i q^{88} +10.0000 q^{89} -8.00000i q^{92} -6.00000i q^{97} -7.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4}+O(q^{10}) \) 2 * q - 2 * q^4	\( 2 q - 2 q^{4} - 8 q^{11} + 2 q^{16} - 8 q^{19} - 2 q^{26} + 12 q^{29} - 16 q^{31} + 12 q^{34} + 12 q^{41} + 8 q^{44} + 16 q^{46} + 14 q^{49} + 8 q^{59} - 4 q^{61} - 2 q^{64} - 32 q^{71} - 20 q^{74} + 8 q^{76} + 32 q^{79} - 8 q^{86} + 20 q^{89}+O(q^{100}) \) 2 * q - 2 * q^4 - 8 * q^11 + 2 * q^16 - 8 * q^19 - 2 * q^26 + 12 * q^29 - 16 * q^31 + 12 * q^34 + 12 * q^41 + 8 * q^44 + 16 * q^46 + 14 * q^49 + 8 * q^59 - 4 * q^61 - 2 * q^64 - 32 * q^71 - 20 * q^74 + 8 * q^76 + 32 * q^79 - 8 * q^86 + 20 * q^89

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